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Kinetics of Monoclonal Antibody Aggregation from Dilute toward Concentrated Conditions Lucrèce Nicoud,† Jakub Jagielski,† David Pfister,† Stefano Lazzari,‡ Jan Massant,§ Marco Lattuada,∥ and Massimo Morbidelli*,† †

Department of Chemistry and Applied Biosciences, ETH Zurich, CH-8093 Zurich, Switzerland Department of Chemical Engineering, MIT, Cambridge, Massachusetts 02139, United States § UCB Pharma, Braine l’Alleud, 1070 Anderlecht, Belgium ∥ Adolphe Merkle Institute, University of Fribourg, 1700 Fribourg, Switzerland ‡

S Supporting Information *

ABSTRACT: Gaining understanding on the aggregation behavior of proteins under concentrated conditions is of both fundamental and industrial relevance. Here, we study the aggregation kinetics of a model monoclonal antibody (mAb) under thermal stress over a wide range of protein concentrations in various buffer solutions. We follow experimentally the monomer depletion and the aggregate growth by size exclusion chromatography with inline light scattering. We describe the experimental results in the frame of a kinetic model based on population balance equations, which allows one to discriminate the contributions of the conformational and of the colloidal stabilities to the global aggregation rate. Finally, we propose an expression for the aggregation rate constant, which accounts for solution viscosity, protein−protein interactions, as well as aggregate compactness. All these effects can be quantified by light scattering techniques. It is found that the model describes well the experimental data under dilute conditions. Under concentrated conditions, good model predictions are obtained when the solution pH is far below the isoelectric point (pI) of the mAb. However, peculiar effects arise when the solution pH is increased toward the mAb pI, and possible explanations are discussed.

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INTRODUCTION Improving our knowledge on protein stability under concentrated conditions is of fundamental relevance in several research fields, such as cell biology1 or biomedical science,2,3 and also has practical applications in various industrial sectors, including the manufacture of food products,4 biomaterials,5 and therapeutic proteins.6 In this study, we focus on the aggregation of monoclonal antibodies (mAbs), which have emerged as a major class of pharmaceuticals since the 1980s.7 There is currently a broad consensus that highly concentrated mAb formulations (in the order of 100 g/L) present notable advantages in terms of patience compliance and convenience of delivery, for example, via subcutaneous route, over more diluted formulations. However, such concentrated formulations give rise to a certain number of issues related to the poor protein solubility, the high solution viscosity, and the formation of protein aggregates, which may reduce drug potency or incite unwanted immune-system responses.6,8 In addition, the use of highly concentrated drug formulations poses some analytical challenges.6 For example, dilution to lower concentrations is generally required prior to analysis, what may impact the results of the assays due to the potential reversibility of loose protein aggregates. Since protein concentration critically affects the kinetics of protein aggregation, understanding the impact of mAb © XXXX American Chemical Society

concentration on mAb stability is a key requisite to optimize the final product formulation. Moreover, due to the limited amount of material available during the early stages of drug development, there is a high interest in assessing formulation stability by performing accelerated studies under dilute conditions. Attempts to extrapolate results from these accelerated studies to conditions relevant for storage require to unravel the role of mAb concentration (as well as temperature or any other accelerating factor) on the kinetics of aggregation.9,10 Gaining a deep understanding of protein aggregation mechanisms under concentrated conditions is nevertheless extremely challenging. Indeed, on top of the complex features affecting the protein aggregation behavior under dilute conditions (such as the protein surface nonuniformity, or the strong interconnection between conformational and colloidal protein stabilities), additional intricate phenomena arise at high protein concentration. For example, viscosity effects,11 multibody interactions,12 excluded volume effects,13,14 short-range interactions,15 activity coefficients,16 positional correlations,17 as well as ion binding18 may become of significant importance in concentrated systems. Received: December 2, 2015 Revised: March 10, 2016

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DOI: 10.1021/acs.jpcb.5b11791 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

mAb, as revealed by circular dichroism measurements.23 It was verified that the aggregates produced at high temperature are not significantly impacted by cooling and dilution in the SEC column during the off-line analysis.23 Although aggregation studies were performed under heat stress, such accelerated conditions can provide fundamental scientific insights into aggregation phenomena, which could be relevant also for the long-term storage of therapeutic proteins at lower temperatures. Size Exclusion Chromatography with Inline Light Scattering. Monomer depletion and aggregate growth were monitored by size exclusion chromatography (SEC) coupled with multiangle light scattering (MALS). More details about the analytical procedure, the determination of the monomer content, oligomer content, and aggregate molecular weight are reported in our previous study.23 Aggregate Fractal Dimension. The fractal dimension of aggregates df was determined from the correlation between the weight-average aggregate molecular weight MWAgg and the average aggregate hydrodynamic radius RAgg h according to

One particular challenge is to evaluate the distinct contributions of these nonideal effects on the aggregation rate. For example, an increase in protein concentration leads on the one hand to a larger excluded volume (expected to accelerate bimolecular aggregation of aggregation-prone species), and on the other hand to an increase in the solution viscosity (expected to slow down aggregation).19 The interplay between these effects makes predictions of the trend of the aggregation rate with the protein concentration extremely difficult. In this work, we combine detailed experimental data and kinetic analysis to investigate the heat-induced aggregation of a model monoclonal antibody from low to moderate protein concentration. Kinetic analysis has indeed proven successful in evaluating the reaction rates of the various mechanistic steps (i.e., protein unfolding and aggregation events) that contribute to the global protein aggregation process.20−24 At low protein concentration, we model the aggregate growth with the classical reaction limited cluster aggregation (RLCA) kernel, which has been shown to describe the aggregation kinetics of a wide range of colloidal systems, and proteins in particular.25,26 After determining relevant model parameters under the ideal case of dilute conditions, we increase the system complexity by progressively moving toward concentrated conditions. At moderate protein concentration, we propose to modify this aggregation rate constant by including two effects: (i) solution nonidealities due to protein−protein interactions, which arise even under stable conditions, and (ii) the increase in solution viscosity during aggregation. Experiments were carried out with four buffer solutions, with different pH values and salt content, with a view to studying the impact of protein−protein interactions on the aggregation rate.

⎛ ⟨R Agg⟩ ⎞d f ⟨MW Agg⟩ = k f ⎜⎜ h ⎟⎟ MWp ⎝ Rp ⎠

(1)

where MWp = 150 kDa and Rp = 6 nm are the molecular weight and the hydrodynamic radius of the monomeric protein, respectively. The prefactor kf is usually in the order of unity.27 To do so, aggregate samples of different sizes were produced by heating protein samples for various incubation times. They were then analyzed with inline light scattering after eluting in the SEC column, as further detailed elsewhere.11 Viscosity Measurement. The viscosity increase during protein aggregation has been monitored in situ at 70 °C by measuring the diffusion coefficient of polymeric tracer nanoparticles with dynamic light scattering (DLS), both in the solvent (+t,0 ) and in the concentrated protein solution (+t ) according to11

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MATERIALS AND METHODS Sample Preparation. The model monoclonal antibody used in this work was a glycosylated IgG1 provided by UCB Pharma, which will be denoted mAb-1 in the following, in agreement with the notation used in a previous study.23 The isoeletric point (pI) of the antibody was determined experimentally both by isoelectric focusing (using an ICE3 system from ProteinSimple at the protein concentration of 2 g/L) and by pH titration (using the Wyatt Mobius system at the protein concentration of 10 g/ L). The two measurements yielded values of 8.7−8.8 and 8.1− 8.2, respectively. The antibody solution was stored at 5−8 °C at protein concentration of 70 g/L in a buffer solution of 20 mM histidine at pH 6.5 containing 250 mM sorbitol. The antibody stock solution contained less than one weight percent of dimer. Prior to aggregation studies, the stock solution was dialyzed against a 20 mM histidine buffer at selected pH (5.5, 6.5 or 7.4) as described elsewhere.23 The samples were then prepared by diluting the dialyzed solution with appropriate buffer solutions to reach the targeted protein concentration (and NaCl concentration in case salt was added) in 20 mM histidine buffers at selected pH values. The pH range used for this study (5.5−7.4) corresponds to the effective pH range of the histidine buffer. All the chemicals were purchased from Sigma, with the highest purity available. The buffers were filtered through a 0.45 μm cutoff membrane filter (Millipore). Isothermal Aggregation Kinetics. Isothermal aggregation kinetics were performed by incubating antibody samples at 70 °C as described in our previous work.23 It is worth mentioning that notable changes in the protein structure occur at the temperature of 70 °C, which is approximately the melting temperature of the

η∞ = η0

+t,0 +t

(2)

where η0 is the solvent viscosity and η∞ is the macroscopic (i.e., felt by “infinitely” large tracer particles) viscosity of the concentrated protein solution. This technique has been shown to provide viscosity values similar to those obtained by traditional rheological measurements provided that the tracer particles are sufficiently large compared to the surrounding protein molecules and stable in the protein solution.11,28 Protein−Protein Interactions. Dynamic Light Scattering. Protein−protein interactions were investigated under stable conditions (i.e., at temperatures which do not induce aggregation during the time frame of the experiments) by DLS using a Zetasizer Nano (Malvern, Worcestershire, U.K.) at fixed angle of 173°. To do so, the mAb collective diffusion coefficient Dc was measured at various protein concentrations c ranging from 1 to 60 g/L. The interaction parameters kD and k′D were then estimated from the following expansion: +c = +0(1 + kDc + kD′ c 2)

(3)

Note that such experiments are typically conducted under dilute conditions (up to about 10 g/L) in order to access to the value of the interaction parameter kD. In this study, a higher order B


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In the limit where the protein concentration approaches 0, g(r) ̅ can be replaced by the expression of exp(−VT(r)/kBT) in the dilute regime, and thus G̅ (c → 0) = −2B̅ . A local Taylor series expansion was computed over small concentration windows to determine the dependence of G = G̅ Na/MWp on the protein concentration:

term was included to extend the description of the diffusion coefficient to the concentrated regime. Experiments were performed at several temperatures, ranging from 25 to 50 °C, which was determined as the highest temperature where aggregation does not occur during the time frame of the experiment. Average values obtained from at least two independent repetitions were considered. Static Light Scattering. To complement the results obtained by DLS, protein−protein interactions between monomeric species were also assessed with static light scattering (SLS) by using a BI-200SM goniometer (Brookhaven Instruments, Holtsville, NY). A solid-state laser Ventus LP532 (Laser Quantum Manchester, U.K.), with a wavelength of λ0 = 532 nm, was used as a light source. The scattered intensity Is was measured at the angle of 90° for a series of samples at various protein concentrations c, ranging from 1 to 65 g/L. The excess Rayleigh ratio R90 was then computed according to R 90

2 R tol ⎛ n ⎞ = (Is − I0) ⎜ ⎟ Itol ⎝ ntol ⎠

R 90 K × MWapp

(4)

(5)

where MWapp is the apparent molecular weight of the protein, B has the unit of a volume divided by a mass, while K is an optical constant defined as ⎛ dn ⎞ 2 K = 4π 2n2⎜ ⎟ Na −1λ 0−4 ⎝ dc ⎠

Figure 1. Scheme of the aggregation mechanism of mAb-1.

First, a native conformation of the monomer N unfolds with a rate constant kU to form an unfolded conformation U which is aggregation-prone due to the exposure of its hydrophobic patches to the solvent. Dimers are then formed and grow irreversibly, either by monomer addition, or by cluster−cluster aggregation. The assumption of irreversible aggregation has been verified by performing dilution experiments on aggregated samples at pH 6.5 in our previous study23 and at other pH values in this work (data not shown). It is worth mentioning that protein unfolding has been shown to be the rate limiting step for monomer depletion under the temperature conditions used in this study. However, it is expected that the unfolding rate significantly decreases when the temperature is lowered toward conditions relevant for long-term storage, possibly leading to a native aggregation mechanism. The population balance equations (PBE) corresponding to the kinetic scheme described above read:

(6)

where Na is the Avogadro number and dn/dc is the sample refractive index increment, which was set to 1.90, in agreement with literature values.30 Formally, the second virial coefficient is defined as the integral over space of the orientation-averaged protein−protein interaction potential:31 B̅ = −2π

∫0

∞⎡

⎤ ⎛ − V (r ) ⎞ ⎢exp⎜ T ⎟ − 1⎥r 2 dr ⎢⎣ ⎝ kBT ⎠ ⎥⎦

(7)

where B̅ = B × MWp/Na has the unit of a volume, kB is the Boltzmann constant, T is the temperature, VT is the total protein−protein interaction potential that accounts for various types of contributions (such as hard sphere, electrostatics, van der Waals, or hydration forces), and r denotes the distance between the protein centers of mass. Under nondilute conditions, protein−protein interactions were examined following the method proposed by Blanco and co-workers.13 Briefly, the Kirkwood−Buff integral G̅ was defined as the orientation-averaged protein−protein correlation function g(r): ̅ G̅ = 4π

∫0

∞

(9)

Protein concentrations ranging from 1 to 10 g/L were used to determine the values of the second virial coefficient B and of the apparent molecular weight MWapp from the Debye plot (eq 5). Protein concentrations up to 65 g/L were used to determine G values with the local Taylor series approach (eq 9). The values of MWapp determined at low protein concentration for each buffer solution were used in eq 9. SLS measurements were performed at the temperature of 50 °C, which was determined as the highest temperature where aggregation does not occur during the time frame of the experiment, with at least two independent repetitions per condition. Aggregation Mechanism. The aggregation mechanism of mAb-1 under thermal stress has already been identified at pH 6.5 under dilute conditions in a previous work,23 as summarized in Figure 1.

where n and ntol represent the refractive index of water and toluene, respectively, equal to n = 1.333 and ntol = 1.496. Moreover, I0 and Itol denote the intensity of the solvent and of toluene, respectively. Rtol is the Rayleigh ratio of toluene, which was set to 1.98 × 10−5 cm−1.29 Under dilute conditions, the protein−protein interactions were studied through the measure of the second virial coefficient B by constructing a Debye plot: Kc 1 = (1 + 2Bc) R 90 MWapp

= c(1 + G(c)c)

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪i ≥ 2 ⎪ ⎩

2

(g ̅ (r ) − 1)r dr

(8) C

dN = −kU N dt ∞

dU = +kU N − U ∑ k1, jUj dt j=1 dUi 1 = dt 2

i−1

∞

∑ kj ,i− jUU j i − j − Ui ∑ k i , jUj j=1

j=1

(10)


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In principle, the Fuchs stability ratio between two monomeric proteins can be computed according to the total protein−protein interaction potential VT according to23,35

It is worth mentioning that a discretization technique based on Gaussian basis functions has been employed in order to reduce the computational time required to solve the PBE shown in eq 10.32−34 Aggregation Rate Constant under Dilute Conditions. The description of the aggregation steps requires the definition of the aggregation rate constant ki,j between two colliding aggregates containing i and j monomeric units. At low protein concentration, and under diffusion limited conditions, the aggregation rate constant of colloidal systems can be described by the following equation:35 ki , j = 4π (+0, i + +0, j)(R c, i + R c, j)

kBT 6πη0R h, i

(11)

(12)

where Rh,i is the hydrodynamic radius of the cluster of mass i. In the frame of the fractal geometry, the collision and hydrodynamic radii can be computed according to Rc,i ≈ Rh,i ≈ i1/df/kf. Under reaction limited conditions, two additional terms must be added to describe the aggregation rate constant under dilute conditions:25,26 k 1 ki , j = S (i1/ d f + j1/ d f )(i−1/ d f + j−1/ d f )(ij)λ 4 Wi , j

+s =

8kBT 3η0

(15)

η0 ⎛ ∂Π ⎞−1 RT ⎟ +c⎜ 2 (1 − cvsp) ηeff (1, 0) ⎝ ∂c ⎠

(16)

where vsp denotes the protein partial specific volume, so that cvsp is equal to the volume fraction occupied by the protein. Moreover, ηeff(1,0) is the effective viscosity experienced by the monomeric protein molecules at time 0, as further described in the following. Interestingly, eq 16 relates the collective diffusion coefficient +c (which quantifies the diffusion of solutes under a concentration gradient, and is given by eq 3) to the self-diffusion coefficient +s (which corresponds to the random motion of one tracer in a colloidal dispersion). Under dilute conditions, the collective and self-diffusion coefficients are equal to the Stokes− Einstein diffusion coefficient +0 given by eq 12. However, under concentrated conditions, nonidealities arise, implying that +c and +s differ from +0 .13,38,39 The derivative of the osmotic pressure with respect to the protein concentration is given by40

(13)

where kS is the Smoluchowski rate constant defined as

kS =

∫2R

However, the so obtained value of the Fuchs stability ratio tends to be extremely sensitive to small uncertainties in the surface charge or potential used to compute VT, and is thus more accurately estimated by fitting model simulations to experimental aggregation kinetic data. Aggregation Rate Constant under Concentrated Conditions. Solution Nonidealities under Stable Conditions. At high mAb concentration, protein−protein interactions affect the rate of protein diffusion, and significant deviations from eq 12 may be observed, implying that the concentration dependent self-diffusion coefficient must be used in eq 11 instead of +0 . By using Stefan−Maxwell equations for multicomponent diffusion, we derived an expression for the self-diffusion coefficient of a monomeric protein solution valid under concentrated conditions.36,37 As compared to the Fick’s law for diffusion, Stefan− Maxwell equations indeed allow to account for thermodynamic nonidealities as well as for the friction between the diffusing species. Details of the derivation are given in the Supporting Information, while the final result is reported here:

where +0, i and Rc,i represent the diffusion coefficient and the collision radius of the cluster containing i monomers, respectively. It must be precised that +0, i corresponds to the diffusion coefficient of single cluster undergoing Brownian motion at infinite dilution. It can be computed from the Stokes− Einstein equation according to +0, i =

⎛ V (r ) ⎞ d r exp⎜ T ⎟ 2 ⎝ kBT ⎠ r p

∞

W11 = 2R p

(14) λ

The term (ij) in eq 13, with λ ≈ 1−1/df, accounts for the number of primary particles located on the external surface of the cluster. Moreover, the factor Wi,j is the so-called Fuchs stability ratio which quantifies the energy barrier that clusters must overcome in order to aggregate. It describes the sticking efficiency with respect to diffusion limited conditions, where all collisions lead to the formation of a larger aggregate. In the case of traditional polymer colloids, it is commonly assumed that a single Fuchs stability ratio (corresponding to the collision between two monomers) can describe the interactions between two clusters, irrespectively of their size.26 However, it is unlikely that this assumption holds true in the case of protein aggregation, where reactivity strongly depends on the accessibility of hydrophobic patches. In particular, U is an unstable intermediate which has a very high reactivity as compared to aggregate species whose hydrophobic patches are already partly covered. Therefore, three types of aggregation events can be identified: monomer−monomer, monomer−aggregate, and aggregate-aggregate. Accordingly, three Fuchs stability ratios are defined: W11, W1j, and Wij, which describe dimer formation, aggregate growth by monomer addition, and aggregate growth by cluster−cluster aggregation, respectively. Depending on the aggregation event considered, the parameter Wi,j in eq 13 takes the value of W11, W1j, or Wij.

∂Π RT = ∂c 1 + cG

(17)

By combining eqs 3, 16, and 17, the self-diffusion coefficient +s can be rewritten as follows: +s = +0 × α

(18)

with α = (1 + kDc + kD′ c 2)(1 + Gc)

η0 1 2 (1 − cvsp) ηeff (1, 0) (19)

Viscosity Increase during Aggregation. When aggregation occurs under concentrated conditions, the diffusion of clusters is affected not only by protein−protein interactions, but also by the increase in the macroscopic viscosity of the solution with time, which slows down the aggregation process. D


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due to protein−protein interactions, as described in the previous section. Proposed Aggregation Kernel. The aggregation rate constant ki,j given in eq 13 for two aggregates of size i and j can now be modified to account for protein−protein interactions and for the increase in the solution viscosity during aggregation. To do so, it is necessary to discuss the size dependence of the correction factor introduced in eq 19. It has been shown previously with DLS experiments that the interaction coefficient kD is independent of the aggregate size.11 Moreover, it was shown that this implies that B Agg /B̅ = MW Agg /MWp = MWAgg/MWp,

The effective viscosity experienced by a given aggregate depends both on the time as well as on the aggregate size. Indeed, “sufficiently large” aggregates feel the macroscopic viscosity (which increases in time due to the increase in the occupied volume fraction11), while “sufficiently small” aggregates rather experience the solvent viscosity. Based on Brownian dynamics simulations performed on concentrated colloidal dispersions stabilized by short-range repulsion, it was shown that the crossover between these two regimes occurs at a size comparable to the average size of the crowders and that the effective viscosity can be computed as follows:34 ηeff (i , t ) = ηnorm(i , t )(η∞(t ) − η0) + η0

where B Agg and MWAgg denote the second virial coefficient and the molecular weight of the aggregates, respectively. Since B = B̅ Na /WWp and BAgg = BAgg Na /MW Agg , it results that BAgg = B, implying that the mass second virial coefficient is independent of the aggregate size (at least in the case under investigation). Since G and B are equivalent under dilute conditions, it is assumed that G is independent of the aggregate size, hypothesis that would need to be verified for other systems. On the other hand, the effective viscosity has been shown to depend on the aggregate size.34 Therefore, the self-diffusion coefficient of an aggregate of size i can be written as +s, i = +0, iαηeff (1, 0)/ηeff (i , t ). One can easily verify that the latter equation simplifies to eq 18 in the case of a stable monomeric protein solution. Finally, by introducing the self-diffusion coefficient in the diffusion term of the classical RLCA kernel, the following kernel is proposed to describe protein aggregation kinetics under concentrated conditions:

(20)

where the normalized viscosity ηnorm is a function comprised between 0 and 1 given by 2/ d f ⎞ ⎛ ⎛ i ⎞ ⎟ ⎜ ηnorm(i , t ) = 1 − exp⎜ −2⎜ ⎟ ⎟ ⎝ ⎝ n w (t ) ⎠ ⎠

(21)

where nw is the dimensionless weight-average mass, defined as the ratio between the second and the first order moment of the aggregate distribution. As illustrated schematically in Figure 2, the scaling proposed above implies that aggregates much larger than the average

ki , j(t ) = α

η (1, 0) ⎞ k S ⎛ −1/ d f ηeff (1, 0) ⎜⎜i + j−1/ d f eff ⎟⎟ ηeff (i , t ) ηeff (j , t ) ⎠ 4⎝

(i1/ d f + j1/ d f ) Figure 2. Viscosity experienced by an aggregate of given size in a protein solution undergoing aggregation.

(23)

where α is computed from eq 19, and ηeff from eqs 20−22. It can be noted that, at low protein concentrations, α → 1 and η∞(t) → η0, so that eq 23 reduces to eq 13.

aggregate size experience the macroscopic solution viscosity (η∞, which increases with time), while aggregates much smaller than the average aggregate size experience a viscosity close to that of the solvent (η0). The aggregates of intermediate sizes feel an effective viscosity comprised between η0 and η∞, as defined in eq 21. The experimental values of the macroscopic viscosity, estimated from the in situ measurement of the diffusion coefficient of tracer particles with DLS (see Materials and Methods section), were given as inputs for the model simulations. To do so, the time evolution of the macroscopic viscosity was fitted for each experimental condition investigated in this work by the following empirical function: ⎛ ⎛ t ⎞ ⎞ η∞(t ) = η0⎜⎜exp⎜ − C2⎟ + C3⎟⎟ ⎠ ⎝ ⎝ C1 ⎠

(ij)λ Wi , j

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RESULTS Aggregation studies were performed at 70 °C in a 20 mM histidine buffer. In order to vary the protein net charge, experiments were carried out at three different pH values (5.5, 6.5, and 7.4) below the protein pI. Moreover, in the case of pH 5.5, an additional experiment was performed in the presence of 10 mM NaCl with a view to assessing the impact of charge screening on the aggregation kinetics. For each of the investigated conditions, experiments were performed at four different mAb concentrations (1, 20, 40, and 60 g/L) and analyzed in the frame of the kinetic model presented above. We discuss first the results obtained under dilute conditions (i.e., at protein concentration of 1 g/L), and then the results obtained under concentrated conditions (i.e., in the range 20−60 g/L). Aggregation under Dilute Conditions. The time evolution of the concentrations of monomer, dimer, and trimer followed by SEC are shown in Figure 3a−c for the four buffer conditions under investigation. Moreover, the increase in the average aggregate molecular weight monitored by inline light scattering after elution in the SEC column is presented in Figure 3d. It is observed that the aggregation kinetics of mAb-1 is

(22)

where C1, C2, and C3 are three positive real numbers, which are reported in Table S1 of the Supporting Information. It is worth underlining that the effective viscosity computed from the combination of eqs 20−22 corresponds to the effective viscosity experienced by a tracer particle interacting by hard-core repulsion only. In order to describe the diffusion of protein molecules, it is necessary to account for solution nonidealities E


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Figure 3. Kinetics of aggregation of mAb-1 at protein concentration of 1 g/L under various buffer conditions, as indicated in the legend. Model simulations (lines) are compared to experimental data (symbols) in terms of (a) monomer depletion, (b) dimer concentration, (c) trimer concentration, and (d) aggregate weight-average molecular weight.

Figure 4. Model parameters at protein concentration of 1 g/L under various buffer conditions. (a) Correlation between aggregate mass and aggregate radius determined by light scattering experiments. Lines correspond to the power law fitting of experimental data. (b) Aggregate fractal dimension determined by regressing the data shown in (a) with eq 1. Error bars represent the 90% confidence interval for the parameter estimation from the regression. (c) Fitted unfolding rate constant as a function of buffer pH. Error bars correspond to the 95% confidence intervals. (d) Fitted Fuchs stability ratios as a function of buffer pH. Open symbols correspond to data obtained at pH 5.5 in the presence of 10 mM NaCl. The 95% confidence intervals are in the order of the symbols’ size.

imental results were interpreted within the kinetic scheme presented in Figure 1, using the aggregation kernel shown in eq 13. The implementation of this kinetic model required the estimation of some parameters, which were determined as follows: (i) the fractal dimension df was assessed from light scattering experiments as shown in Figure 4a, (ii) the exponent λ

strongly affected by the solution conditions. In particular, aggregate growth is found to be much faster at pH 7.4 as compared to the other conditions (Figure 3d). In order to quantify the impact of buffer conditions on the elementary steps that contribute to the global aggregation rate (i.e., protein unfolding and aggregation events), these experF


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Figure 5. Monomer depletion kinetics of mAb-1 at various protein concentrations and in several solution conditions. Experimental data (symbols) are compared to model simulations (lines). The results have been normalized by the initial protein concentration.

in the aggregation kernel was approximated by λ = 1 − 1/df, and (iii) the remaining parameters (i.e., kU, W11, W1j, Wij) were estimated from the fitting of model simulations (see population balance equations shown in eq 10) to the set of experimental data shown in Figure 3. The estimated parameter values are reported in Figure 4b−d, whereas the agreement between model simulations and experimental results can be appreciated in Figure 3. In Figure 4b, it is observed that the aggregate fractal dimension increases when the solution pH increases toward the pI of the mAb, thus indicating that more compact aggregates are formed when the protein net charge is reduced. Interestingly, the addition of 10 mM NaCl at pH 5.5 also leads to an increase in the aggregate fractal dimension. These data suggest that a reduction in the electrostatic repulsion (induced either by a change in pH or by charge screening due to the addition of salt) leads to the formation of denser aggregates, possibly due to the fact that the interpenetration of clusters is favored when the electrostatic barrier is reduced. The values of the unfolding rate constant, which quantifies the protein conformational stability, are shown in Figure 4c as a function of the solution pH. It is seen that the highest value of the unfolding rate constant is obtained at pH 5.5. In order to interpret these results, it is worth recalling that the kinetics of protein unfolding is strongly affected by protein intramolecular and protein−solvent interactions. In particular, it is expected that the protein structure is destabilized at pH values far from the mAb pI due to charge repulsion within the protein molecule,41 as it is observed at pH 5.5. In addition to these nonspecific electrostatic effects, specific charge interactions such as ion pairing between oppositely charged residues can also affect the protein stability.41 These specific electrostatic effects may lead to a stabilization of the folded state at higher protein charge density. This might explain why kU is slightly larger when charge screening is induced by the addition of 10 mM NaCl at pH 5.5. While the protein conformational stability is reflected by the value of the unfolding rate constant, the protein colloidal stability

is described by the value of the Fuchs stability ratio, which quantifies the energy barrier that protein molecules need to overcome before aggregating. As explained in the section dedicated to the kinetic model, three values of the Fuchs stability ratios are required to describe the aggregation of mAb-1: W11, W1j, and Wij, which characterize monomer−monomer, monomer-cluster and cluster−cluster aggregation, respectively. It is observed in Figure 4d that the ranking between the three W values is similar for all the investigated conditions, that is, W11 < W1j < Wij. This ranking can be explained by considering the high reactivity of the unfolded monomer as compared to the aggregates, whose aggregation-prone regions are already partially covered. It is worth noticing that the concentration of dimer is significantly lower at pH 6.5 and 7.4 as compared to pH 5.5 (Figure 3b), suggesting a higher reactivity of the dimer species at higher pH values. This observation was reflected in the kinetic model by setting the value of W12 equal to the value of W11 (instead of W1j) at pH 6.5 and 7.4. In Figure 4d, it is seen that the three Fuchs stability ratio values follow a monotonic decrease with the solution pH. This decrease in colloidal stability can be attributed to the decrease of the protein net charge when the solution pH is increased toward the pI of the protein. Moreover, it is observed that the presence of 10 mM NaCl leads to lower W values at pH 5.5 as compared to the situation in the absence of salt. This effect can be explained by the screening of the protein charge in the presence of salt. To summarize, by combining experimental data with kinetic modeling, we could discriminate the impact of pH on the conformational and colloidal stabilities of the mAb under investigation. While decreasing the pH below the pI destabilizes the protein structure (i.e., promotes monomolecular protein unfolding), it increases protein intermolecular repulsion (i.e., reduces bimolecular protein aggregation). These two competing effects lead to an optimal pH value for which monomer depletion occurs at the slowest rate. It is indeed seen in Figure 3a that monomer depletion is slower at pH 6.5 as compared to pH 5.5 and 7.4. G


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Figure 6. Aggregate growth kinetics of mAb-1 at various protein concentrations and in several solution conditions. Experimental data (symbols) are compared to model simulations (lines).

Figure 7. Kinetics of the increase in solution viscosity followed by measuring the diffusion coefficient of tracer nanoparticles with DLS at various protein concentrations and in several buffer solutions. Symbols correspond to experimental data, and lines to fittings with eq 22.

Aggregation under Concentrated Conditions. In this second part, we move progressively toward higher protein concentrations (i.e., in the range 20−60 g/L) for the same four buffer solution conditions. As under dilute conditions, the kinetics of aggregation was first followed experimentally by SECMALS. The results of the monomer depletion kinetics are shown in Figure 5. To allow a better assessment of the impact of protein concentration on the rate of monomer depletion, the data were normalized by the initial protein concentration. Since monomer depletion has been shown to be rate-limited by monomeric

protein unfolding, these normalized data are expected to overlap on one single curve for each of the investigated solution condition. It can be observed that the concentration profile of monomer obtained at 1 g/L is systematically above those obtained at the larger protein concentrations. This can be attributed to the accumulation of U at low protein concentration.23 SEC analysis indeed gives access only to the sum of the concentrations of the native and unfolded conformations. When the protein concentration increases, the concentration of U becomes negligible, since its disappearance by dimer formation (bimolecular process) is accelerated as compared to its formation H


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The Journal of Physical Chemistry B by protein unfolding (monomolecular process). Therefore, the monomer depletion data acquired at different concentrations are expected to be superimposed when normalized by the initial protein concentration. Such overlapping is observed at pH 5.5 (both in the absence and in the presence of salt), while slight deviations from this ideal behavior are observed at pH 6.5 and 7.4. Possible reasons for this observation will be discussed later on. The increase of the aggregate molecular weight with time is shown in Figure 6 for the four buffer solutions under investigation. It is observed that an increase in protein concentration significantly accelerates the rate of aggregate growth. In addition to monitoring the kinetics of monomer depletion and aggregate growth, the increase in the solution viscosity was followed under concentrated conditions. To do so, the diffusion coefficient of tracer nanoparticles was measured with DLS. Results are presented in Figure 7, where it can be seen that an increase in protein concentration significantly speeds up the rate of viscosity increase. This effect is attributed to the increase of the occupied volume fraction, as demonstrated previously.11 Similarly to the approach used under dilute conditions, we analyzed these data in the frame of the kinetic mechanism presented in Figure 1. As compared to the dilute case, the aggregation kernel shown in eq 23 was used, which accounts for both protein−protein interactions and the increase in viscosity with time. The model parameters were estimated as follows: (i) the fractal dimension was evaluated by light scattering from the mass scaling between aggregate mass and aggregate size shown in eq 1; (ii) the exponent λ in the aggregation kernel was approximated by λ = 1 − 1/df; (iii) the increase in the solution macroscopic viscosity shown in Figure 7 was used to compute the effective viscosity appearing in the aggregation kernel given by eq 23; (iv) the factor α that accounts for protein−protein interactions was evaluated at each mAb concentration and at each buffer conditions through eq 19 by using DLS (to evaluate kD and k′D) and SLS (to evaluate G); (v) the unfolding rate constant kU, which characterizes a monomolecular event, is expected to be independent of protein concentration and was thus set at the value estimated under dilute conditions (shown in Figure 4c); (vi) the Fuchs stability ratios W11, W1j, and Wij were set to the values evaluated under dilute conditions (shown in Figure 4d). It was indeed assumed that the energy barrier that colliding protein molecules must overcome to aggregate once they are at very short distances is independent of the protein concentration. It is worth stressing that, by using this approach, there are no fitting parameters. All model parameters are indeed estimated experimentally or assumed to be equal to those assessed under dilute conditions. The aggregate fractal dimension is plotted as a function of protein concentration in Figure 8a. It is observed that, for all the conditions under investigation, df increases with protein concentration, indicating that denser aggregates are formed. One possible reason is that in concentrated systems, the colliding clusters are likely to be partially overlapped, thus favoring contact in the core of the clusters rather than at their tips. The impact of the initial volume fraction on the fractal dimension of aggregates has been studied by means of Monte Carlo simulations by Gonzáles et al. under DLCA conditions.42 They report a squareroot type function which is plotted as a dashed line in Figure 8a. Interestingly, the experimental data follow a similar trend, with

Figure 8. (a) Aggregate fractal dimension as a function of protein concentration. Dashed line corresponds to the correlation proposed by Gonzáles et al.42 (b) Prefactor of the fractal scaling as a function of the fractal dimension. Dashed line corresponds to the correlation proposed by Ehrl et al.27 Error bars represent the 90% confidence interval for the parameter estimation from the regression shown in eq 1.

values at low protein concentration that depend on the buffer conditions (Figure 4b). For the sake of completeness, the values of the prefactor kf obtained by regressing the light scattering data with eq 1 are plotted as a function of the fractal dimension in Figure 8b. It is observed that kf values are close to unity and tend to decrease with increasing df values, in agreement with the empirical correlation proposed by Ehrl et al.,27 which is represented as a dashed line in Figure 8b. With a view to estimating the parameter α defined in eq 19, protein−protein interactions were quantified with light scattering techniques. It is worth highlighting that the parameter α needs to be estimated under conditions where aggregation does not occur during the time frame of light scattering experiments. For this reason, α was measured at 50 °C, which is lower than the temperature at which aggregation is studied (70 °C). Figure 9a shows the collective diffusion coefficient measured with DLS as a function of the protein concentration. At infinite dilution, hydrodynamic radii close to 6 nm were obtained, slightly decreasing with increasing pH values. The data of Figure 9a were regressed with eq 3 in order to obtain the interaction parameters kD and k′D, and the corresponding fits are shown by lines in the same figure. The so obtained kD values are shown in Figure 9b, which also contains data obtained at temperatures lower than 50 °C in order to assess the temperature dependence of the kD parameter. It can be seen in Figure 9b that kD is positive and decreases upon the addition of NaCl. This indicates a decrease of the strength of repulsive interactions due to charge screening. Moreover, it is observed that kD increases when the solution pH increases from 5.5 to 7.4. It is worth noting that an increase in the solution pH toward the mAb pI leads on the one hand to a decrease in the protein net charge, and on the other hand to a I


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Figure 9. Protein−protein interactions from low to moderate protein concentration under various solution conditions. (a) Concentration dependence of the collective diffusion coefficient measured with DLS at 50 °C. Lines correspond to fittings with eq 3. (b) Temperature dependence of the first order interaction coefficient determined with eq 3. Lines, which correspond to linear fits, are guides for the eyes. (c) Concentration dependence of the Rayleigh ratio measured with SLS at 50 °C. (d) Concentration dependence of the Kirkwood−Buff integral at 50 °C determined with eq 9. (e) Correlation between kD and B at 50 °C. Lines correspond to literature correlations: Lehermayr et al.43 (solid line), Connolly et al.44 (dashed line), and Saito et al.45 (dotted line). (f) Correction factor α as a function of protein concentration calculated with eq 19 at 50 °C. Error bars in panels (b), (c), (d), and (e) represent the 80% confidence intervals.

However, it must be realized that the two aforementioned quantities are not sensitive to the same characteristics of the interaction potential, implying that the ranking in B values does not necessarily correspond to the ranking in W11 values (see Figure S3 in the Supporting Information). The SLS data shown in Figure 9c were then regressed over small concentration windows to determine the concentration dependence of the Kirkwood-Buff integral G using eq 9. The obtained results are shown in Figure 9d, where it is seen that for all the tested conditions G is negative and increases with protein concentration, indicating that protein−protein interactions are overall repulsive, and that their strength decreases with protein concentration, as already observed in the literature for another protein system.13 Finally, the values of kD, k′D, and G(c) that were measured at 50 °C for each investigated condition were used to estimate the parameter α defined by eq 19. Results are shown in Figure 9f, where it is seen that α, which represents the factor by which the self-diffusion coefficient is reduced as compared to the dilute case, decreases with the protein concentration with a trend that depends on the solution condition (pH and ionic strength). The model parameter values obtained above can now be used to compute the aggregation kernel (eq 23) and then to predict the kinetics of mAb aggregation under concentrated conditions. The model simulations of the monomer depletion kinetics are compared to the experimental data in Figure 5. It is seen that the model describes well the experimental results at pH 5.5, both in the absence and presence of salt. However, it tends to underestimate the rate of monomer consumption when the pH of the buffer solution is increased toward the protein pI. Two main reasons may underlie these results:

decrease in the ionic strength of the buffer solutions, which is around 15 mM, 5 mM and 1 mM at pH 5.5, 6.5 and 7.4, respectively. Indeed, higher amounts of HCl and of positive charges on the histidine molecules lead to a higher ionic strength at low pH. It was verified that kD correlates with the protein net charge (i.e., decreases when the solution pH is increased toward the mAb pI) when the ionic strength is kept constant at 15 mM by adding 10 and 14 mM NaCl at pH 6.5 and 7.4, respectively (see Figure S1 in the Supporting Information). With regard to the second order interaction parameter, negative values of kD′ are found, with absolute values increasing when the pH is increased toward the mAb pI: −0.08, −0.14, and −0.20 at pH 5.5, 6.5, and 7.4, respectively. At pH 5.5 in the presence of salt, k′D is found to be close to 0. The results of the static light scattering experiments are shown in Figure 9c where it can be seen that the Rayleigh ratio increases with the protein concentration until around 40 g/L, and then flattens out. In the range from 1 to 10 g/L, these data were used to determine the second virial coefficient B and the apparent molecular weight MWapp by using eq 5 (see Figure S2 in the Supporting Information). It was observed that that the so obtained second virial coefficient B (measured by SLS) correlates with the interaction parameter kD (measured by DLS), as shown in Figure 9e. This finding is in good agreement with similar empirical correlations previously reported in the literature,43−45 and represented by the lines in Figure 9e. Moreover, it is interesting to note that the second virial coefficient and the Fuchs stability ratio follow different trends with the solution pH: W11 decreases with the solution pH, while B (as well as kD) increases with the solution pH. This observation might be surprising at first since both B and W11 are defined from the protein interaction potential (see eqs 7 and 15, respectively). J


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the correction factor α arise due to protein−protein interactions. In the following, we discuss how the kind and strength of intermolecular protein−protein interactions are affected by an increase in the protein concentration from dilute to concentrated conditions. First, a crowding or excluded volume ef fect arises at high concentration due to the decrease in the free volume available to protein molecules.13,14 Moreover, since protein molecules are on average closer from each other at higher concentration, both long- and short-range electrostatic interactions are of significant importance under concentrated conditions.15 Increasing the protein concentration also results in an increase in the solution ionic strength due to17,49 (i) the increase in the concentration of counterions necessary to maintain electroneutrality, and (ii) the reduction of the volume available to the counterions.49 In the absence of ion specific effects, an increase in the ionic strength is expected to screen electrostatic interactions. Nonetheless, counterion binding to the protein surface, which possibly becomes relevant at high protein concentration,18 may affect intermolecular interactions in a complex manner.50 It has been reported in the literature that attractive interactions may also be affected by a change in protein concentration.17,51−53 A reduction of the strength of attractive interactions with increasing protein concentration was indeed evidenced with small-angle X-ray scattering studies performed on globular proteins.17,51,52 Nevertheless, this experimental observation, whose underlying reasons are still unclear, could potentially be an artifact due to many-body interactions. Indeed, under concentrated conditions, the pairwise additivity of intermolecular potentials breaks down and many-body interactions become non-negligible. In particular, an attractive three-body interaction was identified, both for a protein system12 and for charged colloidal particles.54 Viscosity. In addition to protein−protein interactions, which affect the rate of protein diffusion even under stable conditions, the rise in solution viscosity during aggregation further hinders protein mobility. The increase in the macroscopic viscosity of the solution is attributed to the increase in the aggregate occupied volume fraction, and can be quantified experimentally, for example from the measure of the diffusion of tracer nanoparticles with DLS.11 The impact of a rise in solution viscosity on the aggregation rate constant depends on the aggregate size: aggregates much larger than the average aggregate size experience the macroscopic viscosity of the dispersion, while aggregates much smaller than the average aggregate size experience the solvent viscosity. The aggregates of intermediate sizes experience an effective viscosity comprised between the solvent and the macroscopic viscosities, as defined in eq 21. Aggregate Compactness. Finally, it was observed with light scattering experiments that more compact aggregates are formed under concentrated conditions. This effect does not require the addition of a special correction term in the aggregation kernel, since the aggregate compactness is already taken into account through the value of the fractal dimension df. A change in aggregate morphology impacts the aggregation rate in a complex way. Indeed, for a given aggregate dimensionless mass i, an increase in the aggregate fractal dimension df leads to a smaller collision radius (∼i1/df), a larger diffusive mobility (∼i−1/df), and a larger number of possible contact points at the external surface of the cluster (∼i1−1/df). While the former contribution leads to a decrease in aggregate

(i) The unfolding rate constant increases with protein concentration because surrounding protein molecules exhibit a destabilizing effect on the protein structure. Although this effect can hardly be corroborated with the set of experimental techniques used in this study, it cannot be ruled out. (ii) A fraction of the monomeric protein aggregates in a native conformation, in addition to the aggregation of unfolded proteins. Experimental and computational studies indeed revealed that molecular crowding may induce a change in the shape of the native protein, potentially leading to the exposure of aggregation-prone patches.46 Moreover, it can be speculated that native monomer aggregation is favored at pH close to the pI of the protein, where electrostatic repulsion is very weak, possibly explaining larger discrepancies between model simulations and experimental data at pH 7.4. Let us now consider the model predictions of the time evolution of the aggregate molecular weight shown in Figure 6. A comparison of these model predictions with model predictions obtained by (i) neglecting solution nonidealities (i.e., considering α = 1 in eq 23), and (ii) neglecting both solution nonidealities and viscosity effects (i.e., using the classical RLCA kernel shown in eq 13) can be found in the Supporting Information. It is seen in Figure 6 that the model describes well the set of experimental data at pH 5.5 and 6.5. The discrepancies between model simulations and experimental results observed at pH 5.5 in the presence of NaCl can still be regarded as acceptable considering that there is no fitting parameter. However, at pH 7.4, the model fails to describe the kinetics of aggregate growth. In particular, the simulated aggregate molecular weight is significantly overestimated at the three protein concentrations under investigation. Two main reasons can be proposed to explain this observation: (i) The strength of protein−protein interactions significantly changes between 50 and 70 °C, implying that the parameter α estimated at 50 °C under stable conditions cannot be used to describe aggregation kinetics at 70 °C. This reasoning is supported by the fact that the kD parameter is strongly impacted by temperature at pH 7.4, as compared to the other solution conditions, as can be seen in Figure 9b. (ii) Protein surface nonuniformity and anisotropic interactions become critical when the solution pH approaches the pI of the protein,47,48 possibly explaining the failure of coarse-grained molecular potentials (quantified through kD and G) to describe orientation dependent protein−protein interactions.

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DISCUSSION Aggregation Kernel under Concentrated Conditions. The classical RLCA kernel derived under dilute conditions accounts for two main features of the aggregating system: the energy barrier that the colliding proteins must overcome before aggregating, and the aggregate compactness. In this work, we propose a new aggregation kernel, as indicated in eq 23, in order to extend the classical expression to the concentrated regime. This model includes two additional contributions as compared to the dilute case: (i) A correction factor α, given by eq 19, reflecting the hydrodynamic and thermodynamic nonidealities of the concentrated protein solution under stable conditions. (ii) A correction term taking into account the viscosity increase during aggregation, as described by eqs 20 and 21. These two effects vanish at low protein concentration, thus reducing the proposed model to the classical RLCA kernel. Protein Solution Nonidealities. The hydrodynamic and thermodynamic solution nonidealities that are quantified by K


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low protein concentration (pH 7.4) does not correspond to the least stable condition at moderate protein concentration (pH 5.5 + 10 mM NaCl). Moreover, it is interesting to note that the destabilizing effect of NaCl is much more pronounced at 60 g/L than at 1 g/L. These experimental observations illustrate the fact that the concentration dependence of the aggregation rate constant depends on the solution conditions. These considerations are particularly relevant in the context of drug formulation since they imply that identifying the most stable formulation under dilute conditions does not guarantee to select the most stable conditions under concentrated conditions.

reactivity, the latter two lead to an increase in aggregate reactivity. Numerical evaluation of the reaction limited aggregation kernel show that, overall, an increase in the aggregate compactness at high concentration leads to an increase in the aggregation rate constant. Stability Ranking from Dilute to Concentrated Conditions. It has to be noted that the relative magnitude of the aforementioned effects on the aggregation rate depends on the specific system under investigation. For example, the increase of the ionic strength with protein concentration may become particularly important when the mAb molecules are highly charged or the background ion concentration is low.49 Moreover, many-body interactions are expected to be stronger at low ionic strength, where double layers overlap.55 Therefore, the concentration dependence of the aggregation rate is likely to be formulation specific, possibly resulting in a change of the stability ranking when moving from dilute to concentrated conditions. This idea is illustrated in Figure 10, where the fraction of residual monomer and the aggregate molecular weights at 15 min

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CONCLUSION We investigated the heat-induced aggregation of a model monoclonal antibody from low to moderate protein concentration under various buffer solution conditions. To do so, we combined experimental characterization with a kinetic model based on population balance equations which includes protein unfolding and aggregation events. Under dilute conditions, the aggregate growth was successfully described by using the classical reaction limited aggregation kernel, which depends on two main factors: the aggregate compactness (quantified by the aggregate fractal dimension) and the energy barrier preventing the colliding protein molecules from aggregating (quantified by the Fuchs stability ratio). It was observed that decreasing the solution pH below the protein pI leads on the one hand to an increase in the colloidal stability (reflected by an increase in the Fuchs stability ratio), and on the other hand to a decrease in the conformational stability (reflected by an increase in the unfolding rate constant). This implies that there is an optimum pH that minimizes the rate of monomer depletion. Under concentrated conditions, we proposed to modify the classical aggregation rate constant to account for two effects: (i) hydrodynamic and thermodynamic solution nonidealities, that can be evaluated under stable conditions with light scattering techniques; and (ii) the rise in solution viscosity during the aggregation process, that can be accounted for by using a size and time dependent effective viscosity. Depending on the aggregate size, this effective viscosity is comprised between the solvent and the macroscopic viscosity of the solution, which can be quantified from the measure of the diffusion of tracer particles with DLS. At pH far below the pI, this model was shown to describe well the set of experimental data, in terms of both monomer depletion and aggregate growth. When the pH was increased toward the mAb pI, discrepancies were observed between model simulations and experimental results. Indeed, model simulations tend to (i) slightly underestimate the rate of monomer depletion, possibly due to an increase in the unfolding rate constant or to native monomer aggregation, and (ii) considerably overestimate the rate of aggregate growth, possibly due to the strong temperature dependence of protein−protein interactions (which makes the estimation of protein−protein interactions under stable conditions non applicable to aggregating conditions at pH close to the pI), or to strong anisotropic interactions (that make coarse-grained molecular potentials fail to describe protein− protein interactions at pH close to the pI). Importantly, the finding that the concentration dependence of the aggregation rate constant is specific to the buffer solution implies that the most stable formulation under dilute conditions is not necessarily the most stable formulation under concentrated conditions.

Figure 10. (a) Fraction of residual monomer and (b) aggregate molecular weight experimentally measured at 15 min incubation as a function of the solution conditions at protein concentrations of 1 and 60 g/L.

incubation measured in the previous sections are plotted as a function of the solution condition, at protein concentrations of 1 and 60 g/L. Considering first the data of the residual monomer (Figure 10a), it is seen that the stability ranking is unchanged when moving from 1 to 60 g/L. This observation is in line with the fact that, in the case under investigation, the kinetics of monomer depletion is rate limited by protein unfolding, which is only weakly affected by a change in protein concentration. Let us consider now the results of the aggregate molecular weight, which is affected by both protein unfolding and bimolecular aggregation. It is seen in Figure 10b that the most stable condition at low protein concentration (pH 5.5) does not correspond to the most stable condition at moderate protein concentration (pH 6.5). Similarly, the least stable condition at L


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b11791. Derivation of the expression for the self-diffusion coefficient under concentrated conditions, values of the parameters C1, C2, and C3 characterizing the viscosity increase measured experimentally during aggregation, zeta potential measurements, interaction parameter at constant ionic strength, Debye plots with corresponding B and MWapp values, correlation between the second virial coefficient and the Fuchs stability ratio, and model predictions neglecting nonideal effects (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Mailing address: Institute for Chemical and Bioengineering, Department of Chemistry & Applied Biosciences, ETH Zurich, Vladimir-Prelog-Weg 1, CH-8093 Zurich. Phone: +41 44 632 30 34. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank UCB Pharma (Braine l’Alleud, Belgium) for material supplying and financial support. Moreover, financial support from the Fondation Claude et Giuliana and from the Swiss National Science Foundation (Grant Numbers 200020_147137/1, PP00P2133597/1, P2EZP2_159128) is gratefully acknowledged.

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